L(s) = 1 | − 0.656i·2-s + 1.56·4-s + 3.31·5-s − 2.34i·8-s − 2.17i·10-s + 2.34i·11-s − 1.58i·13-s + 1.59·16-s + 1.13·17-s + 4.44i·19-s + 5.19·20-s + 1.53·22-s − 9.39i·23-s + 5.96·25-s − 1.03·26-s + ⋯ |
L(s) = 1 | − 0.464i·2-s + 0.784·4-s + 1.48·5-s − 0.828i·8-s − 0.687i·10-s + 0.706i·11-s − 0.438i·13-s + 0.399·16-s + 0.275·17-s + 1.02i·19-s + 1.16·20-s + 0.328·22-s − 1.95i·23-s + 1.19·25-s − 0.203·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.737333788\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.737333788\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.656iT - 2T^{2} \) |
| 5 | \( 1 - 3.31T + 5T^{2} \) |
| 11 | \( 1 - 2.34iT - 11T^{2} \) |
| 13 | \( 1 + 1.58iT - 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 - 4.44iT - 19T^{2} \) |
| 23 | \( 1 + 9.39iT - 23T^{2} \) |
| 29 | \( 1 + 3.65iT - 29T^{2} \) |
| 31 | \( 1 - 7.31iT - 31T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 - 9.64T + 41T^{2} \) |
| 43 | \( 1 + 2.16T + 43T^{2} \) |
| 47 | \( 1 + 5.58T + 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 2.17T + 59T^{2} \) |
| 61 | \( 1 - 4.00iT - 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 6.39iT - 71T^{2} \) |
| 73 | \( 1 - 10.6iT - 73T^{2} \) |
| 79 | \( 1 - 1.23T + 79T^{2} \) |
| 83 | \( 1 + 1.03T + 83T^{2} \) |
| 89 | \( 1 - 7.47T + 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.858161051744876984708242462520, −8.931522114478221334000436651063, −7.892289405685730305549323116854, −6.90132237097931043554206503427, −6.21904360053033021775130461403, −5.53314344682375119632686977674, −4.37023477019508173335943946636, −3.01371979984636755571761968739, −2.22312869077430032336123706928, −1.32996791150708728023108016905,
1.48178921500680867280685967511, 2.39140283669171697157005521038, 3.43301754348880999556572626976, 5.07559520594492684912470380527, 5.70317951900057604290552758170, 6.34944525587853487003650751921, 7.12629595080731366097745989707, 7.975581051692301811328029981081, 9.063110461659109247962790824035, 9.576457071938940324277610300600